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Trapping of null geodesics in slowly rotating extremely compact Tolman VII spacetimes

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Abstract

Region of trapped null geodesics hidden inside of extremely compact objects is of astrophysical importance because of trapping of gravitational waves, or neutrinos. The trapping effect of null geodesics was extensively studied for spherically symmetric extremely compact objects. Recently, influence of rotation of the extremely compact objects on the trapping of null geodesics was treated in the simplest possible model of internal linearized Hartle–Thorne spacetime with uniform energy density distribution and uniform emissivity distribution of null geodesics. Here, we extend the study of the rotation influence on the trapping effect in the case of linearized Hartle–Thorne spacetimes based on the Tolman VII spherically symmetric solutions, where we assume the emissivity of the null geodesics proportional to the energy density of the Tolman VII object having quadratic radial profile. We demonstrate enhancement (suppression) of the trapping effect in the case of counter-rotating (co-rotating) null geodesics due to the behavior of the effective potentials and escape cones of the null geodesics in the linearized Hartle–Thorne–Tolman VII spacetimes. In dependence on the parameters of these spacetimes, we determine the “local” and “global” coefficients of efficiency of the trapping and compare the results to those related to the rotating spacetimes based on the internal Schwarzschild spacetimes. We demonstrate that in the Tolman VII spacetimes the trapping is more efficient, being allowed in objects with radii larger than those of the trapping internal Schwarzschild spacetimes, occurring even for \(R>3.3M\).

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Notes

  1. The special character of the internal Schwarzschild spacetimes in the limit of \(R \rightarrow 2M\) related to gravastars is discussed in [17,18,19].

  2. The extremely compact polytropic spheres can also have extension overcoming the Schwarzschild photosphere radius \(R=3M\) [24,25,26,27,28].

  3. Note that in the rotating spacetimes the symmetry of the trapping zone (cone) is lost as the motion depends on the sign of the impact parameter, as demonstrated for the case of trapped null geodesics in Kerr spacetimes [39, 40].

  4. We follow only procedure not the signature of metric tensor.

  5. In order to indicate possible behavior of the trapping effects for the standard internal Hartle–Thorne–Tolman VII spacetimes, we consider here also relatively large values of the rotation parameter j rising up to \(j=0.5\). Note that \(j=0.7\) is considered to be the limiting value acceptable for the Hartle–Thorne spacetimes taken to the quadratic approximation in the angular velocity \(\Omega \) [48].

References

  1. M.A. Abramowicz, J.C. Miller, Z. Stuchlík, Concept of radius of gyration in general relativity. Phys. Rev. D 47(4), 1440–1447 (1993). https://doi.org/10.1103/PhysRevD.47.1440

    Article  ADS  Google Scholar 

  2. F. Felice, Equatorial geodesic motion in the gravitational field of a rotating source. Nuovo Cimento B Ser. 57(2), 351–388 (1968). https://doi.org/10.1007/BF02710207

    Article  ADS  Google Scholar 

  3. M.A. Abramowicz, N. Andersson, M. Bruni, P. Ghosh, S. Sonego, Letter to the Editor: Gravitational waves from ultracompact stars: the optical geometry view of trapped modes. Class. Quantum Gravity 14(12), L189–L194 (1997). https://doi.org/10.1088/0264-9381/14/12/002

    Article  ADS  MATH  Google Scholar 

  4. Z. Stuchlík, G. Török, S. Hledík, M. Urbanec, Neutrino trapping in extremely compact objects: I. Efficiency of trapping in the internal Schwarzschild spacetimes. Class. Quantum Gravity 26(3), 035003 (2009). https://doi.org/10.1088/0264-9381/26/3/035003

    Article  ADS  MATH  Google Scholar 

  5. L. Barack et al., Black holes, gravitational waves and fundamental physics: a roadmap. Class. Quantum Gravity 36(14), 143001 (2019). https://doi.org/10.1088/1361-6382/ab0587

    Article  ADS  MathSciNet  Google Scholar 

  6. V. Cardoso, A.S. Miranda, E. Berti, H. Witek, V.T. Zanchin, Geodesic stability, Lyapunov exponents, and quasinormal modes. Phys. Rev. D 79(6), 064016 (2009). https://doi.org/10.1103/PhysRevD.79.064016

    Article  ADS  MathSciNet  Google Scholar 

  7. R.A. Konoplya, Z. Stuchlík, Are eikonal quasinormal modes linked to the unstable circular null geodesics? Phys. Lett. B 771, 597–602 (2017). https://doi.org/10.1016/j.physletb.2017.06.015

    Article  ADS  MATH  Google Scholar 

  8. Z. Stuchlík, J. Schee, Shadow of the regular Bardeen black holes and comparison of the motion of photons and neutrinos. Eur. Phys. J. C 79(1), 44 (2019). https://doi.org/10.1140/epjc/s10052-019-6543-8

    Article  ADS  Google Scholar 

  9. B. Toshmatov, Z. Stuchlík, J. Schee, B. Ahmedov, Electromagnetic perturbations of black holes in general relativity coupled to nonlinear electrodynamics. Phys. Rev. D 97(8), 084058 (2018). https://doi.org/10.1103/PhysRevD.97.084058

    Article  ADS  MathSciNet  Google Scholar 

  10. B. Toshmatov, Z. Stuchlík, B. Ahmedov, D. Malafarina, Relaxations of perturbations of spacetimes in general relativity coupled to nonlinear electrodynamics. Phys. Rev. D 99(6), 064043 (2019). https://doi.org/10.1103/PhysRevD.99.064043

    Article  ADS  MathSciNet  Google Scholar 

  11. N.K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics, and General Relativity (Springer, Berlin, 2000)

    Book  Google Scholar 

  12. S.L. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (Wiley, Hoboken, 1986)

    Google Scholar 

  13. F. Weber, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics (Routledge, Abingdon, 1999)

    Google Scholar 

  14. Z. Stuchlík, J. Hladík, M. Urbanec, Neutrino trapping in braneworld extremely compact stars. Gen. Relativ. Gravit. 43, 3163–3190 (2011). https://doi.org/10.1007/s10714-011-1229-z

    Article  ADS  MATH  Google Scholar 

  15. Z. Stuchlík, Influence of the RELICT cosmological constant on accretion discs. Mod. Phys. Lett. A 20, 561–575 (2005). https://doi.org/10.1142/S0217732305016865

    Article  ADS  MATH  Google Scholar 

  16. Z. Stuchlík, M. Kološ, J. Kovář, P. Slaný, A. Tursunov, Influence of cosmic repulsion and magnetic fields on accretion disks rotating around Kerr black holes. Universe 6(2), 26 (2020). https://doi.org/10.3390/universe6020026

    Article  ADS  Google Scholar 

  17. R.A. Konoplya, C. Posada, Z. Stuchlík, A. Zhidenko, Stable Schwarzschild stars as black-hole mimickers. Phys. Rev. D 100(4), 044027 (2019). https://doi.org/10.1103/PhysRevD.100.044027

    Article  ADS  MathSciNet  Google Scholar 

  18. J. Ovalle, C. Posada, Z. Stuchlík, Anisotropic ultracompact Schwarzschild star by gravitational decoupling. Class. Quantum Gravity 36(20), 205010 (2019). https://doi.org/10.1088/1361-6382/ab4461

    Article  ADS  MathSciNet  Google Scholar 

  19. C. Posada, C. Chirenti, On the radial stability of ultra-compact Schwarzschild stars beyond the Buchdahl limit. Class. Quantum Gravity 36(6), 065004 (2019). https://doi.org/10.1088/1361-6382/ab0526

    Article  ADS  MathSciNet  Google Scholar 

  20. L. Randall, R. Sundrum, An alternative to compactification. Phys. Rev. Lett. 83, 4690–4693 (1999). https://doi.org/10.1103/PhysRevLett.83.4690

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Z. Stuchlík, J. Vrba, J. Hladík, C. Posada, Neutrino trapping in extremely compact Tolman VII spacetimes. Eur. Phys. J. C 81, 1–3 (2021)

    Article  Google Scholar 

  22. R.C. Tolman, Static solutions of Einstein’s field equations for spheres of fluids. Phys. Rev. 55, 364 (1939)

    Article  ADS  Google Scholar 

  23. N. Neary, M. Ishak, K. Lake, Tolman type VII solution, trapped null orbits, and w-modes. Phys. Rev. D 64(8), 084001 (2001). https://doi.org/10.1103/PhysRevD.64.084001

    Article  ADS  Google Scholar 

  24. S. Hod, Lower bound on the compactness of isotropic ultracompact objects. Phys. Rev. D 97(8), 084018 (2018). https://doi.org/10.1103/PhysRevD.97.084018

    Article  ADS  MathSciNet  Google Scholar 

  25. J. Novotný, J. Hladík, Z. Stuchlík, Polytropic spheres containing regions of trapped null geodesics. Phys. Rev. D 95(4), 043009 (2017). https://doi.org/10.1103/PhysRevD.95.043009

    Article  ADS  Google Scholar 

  26. Y. Peng, Upper bounds on the compactness at the innermost light ring of anisotropic horizonless spheres. Eur. Phys. J. C 80(8), 755 (2020). https://doi.org/10.1140/epjc/s10052-020-8358-z

    Article  ADS  Google Scholar 

  27. Z. Stuchlík, S. Hledík, J. Novotný, General relativistic polytropes with a repulsive cosmological constant. Phys. Rev. D 94(10), 103513 (2016). https://doi.org/10.1103/PhysRevD.94.103513

    Article  ADS  Google Scholar 

  28. Z. Stuchlík, J. Schee, B. Toshmatov, J. Hladík, J. Novotný, Gravitational instability of polytropic spheres containing region of trapped null geodesics: a possible explanation of central supermassive black holes in galactic halos. JCAP 6, 056 (2017). https://doi.org/10.1088/1475-7516/2017/06/056

    Article  ADS  MathSciNet  Google Scholar 

  29. N. Jiang, K. Yagi, Improved analytic modeling of neutron star interiors. Phys. Rev. D 99(12), 124029 (2019). https://doi.org/10.1103/PhysRevD.99.124029

    Article  ADS  MathSciNet  Google Scholar 

  30. N. Jiang, K. Yagi, Analytic I-Love-C relations for realistic neutron stars. Phys. Rev. D 101(12), 12–4006 (2020). https://doi.org/10.1103/PhysRevD.101.124006

    Article  MathSciNet  Google Scholar 

  31. C. Posada, J. Hladík, Z. Stuchlík, Dynamical stability of the modified Tolman VII solution. arXiv e-prints arXiv:2103.12867 (2021)

  32. S. Hensh, Z. Stuchlík, Anisotropic Tolman VII solution by gravitational decoupling. Eur. Phys. J. C 79(10), 834 (2019). https://doi.org/10.1140/epjc/s10052-019-7360-9

    Article  ADS  Google Scholar 

  33. J.B. Hartle, K.S. Thorne, Slowly rotating relativistic stars. II. Models for neutron stars and supermassive stars. Astrophys. J. 153, 807 (1968). https://doi.org/10.1086/149707

    Article  ADS  Google Scholar 

  34. J. Vrba, M. Urbanec, Z. Stuchlík, J.C. Miller, Trapping of null geodesics in slowly rotating spacetimes. Eur. Phys. J. C 80(11), 1065 (2020). https://doi.org/10.1140/epjc/s10052-020-08642-z

    Article  ADS  Google Scholar 

  35. C.G. Böhmer, Eleven spherically symmetric constant density solutions with cosmological constant. Gen. Relativ. Gravit. 36(5), 1039–1054 (2004). https://doi.org/10.1023/B:GERG.0000018088.69051.3b

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. K. Schwarzschild. On the gravitational field of a mass point according to Einstein’s theory. Abh. Konigl. Preuss. Akad. Wissenschaften Jahre 1906, 92, Berlin, 1907, 189–196 (1916)

  37. Z. Stuchlík, Spherically symmetric static configurations of uniform density in spacetimes with a non-zero cosmological constant. Acta Phys. Slovaca 50(2), 219–228 (2000)

    Google Scholar 

  38. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (1973)

  39. Z. Stuchlík, J. Schee, Appearance of Keplerian discs orbiting Kerr superspinars. Class. Quantum Gravity 27(21), 215017 (2010). https://doi.org/10.1088/0264-9381/27/21/215017

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Z. Stuchlík, D. Charbulák, J. Schee, Light escape cones in local reference frames of Kerr-de Sitter black hole spacetimes and related black hole shadows. Eur. Phys. J. C 78(3), 180 (2018). https://doi.org/10.1140/epjc/s10052-018-5578-6

    Article  ADS  Google Scholar 

  41. J.M. Bardeen, W.H. Press, S.A. Teukolsky, Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation. Astrophys. J. 178, 347–370 (1972). https://doi.org/10.1086/151796

    Article  ADS  Google Scholar 

  42. J.C. Miller, Quasi-stationary gravitational collapse of slowly rotating bodies in general relativity. Mon. Not. R. Astron. Soc. 179, 483–498 (1977). https://doi.org/10.1093/mnras/179.3.483

    Article  ADS  Google Scholar 

  43. M.A. Abramowicz, G.J.E. Almergren, W. Kluzniak, A.V. Thampan, The Hartle–Thorne circular geodesics. arXiv e-prints arXiv:art.gr-qc/0312070, Dec (2003)

  44. S. Chandrasekhar, The Mathematical Theory of Black Holes (Clarendon Press/Oxford University Press, Oxford/New York, 1983)

    MATH  Google Scholar 

  45. J. Schee, Z. Stuchlík, Optical phenomena in the field of braneworld Kerr black holes. Int. J. Mod. Phys. D 18, 983–1024 (2009). https://doi.org/10.1142/S0218271809014881

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Z. Stuchlík, A. Kotrlová, Orbital resonances in discs around braneworld Kerr black holes. Gen. Relativ. Gravitat. 41, 1305–1343 (2009). https://doi.org/10.1007/s10714-008-0709-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Z. Stuchlík, J. Hladík, M. Urbanec, G. Török, Neutrino trapping in extremely compact objects described by the internal Schwarzschild-(anti-)de Sitter spacetimes. Gen. Relativ. Gravit. 44(6), 1393–1417 (2012). https://doi.org/10.1007/s10714-012-1346-3

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. M. Urbanec, J.C. Miller, Z. Stuchlík, Quadrupole moments of rotating neutron stars and strange stars. Mon. Not. RAS 433(3), 1903–1909 (2013). https://doi.org/10.1093/mnras/stt858

    Article  ADS  Google Scholar 

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Acknowledgements

J.V. and Z.S. acknowledge the institutional support of the Institute of Physics, Silesian University in Opava. J.V. was supported by the Czech Grant No. LTC18058.

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  1. Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo nám. 13, 746 01, Opava, Czech Republic

    Zdeněk Stuchlík & Jaroslav Vrba

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Correspondence to Jaroslav Vrba.

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Stuchlík, Z., Vrba, J. Trapping of null geodesics in slowly rotating extremely compact Tolman VII spacetimes. Eur. Phys. J. Plus 136, 977 (2021). https://doi.org/10.1140/epjp/s13360-021-01890-2

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